Prof. Ernst Terhardt

Strike note of bells

The sound of bells, and how it is perceived, is of high significance
for the understanding of auditory perception, in particular, pitch perception.
As was pointed out in the topic definition
of pitch, pitch is a multiple auditory attribute, and the sound
of a bell makes this drastically apparent. A considerable number of spectral
pitches can be "heard out" which correspond to the frequencies of
the bell's eigen-oscillations. Besides these "analytic" auditory aspects
of the sound, there is a "holistic" one, namely, the auditory percept
of a musical pitch that is more or less pronounced and fairly unambiguous
- depending on the quality of the bell. That holistic pitch, which by
the ear is associated with the sound immediately after the strike, has
traditionally been termed the strike note. Explanation of the strike
not is not trivial, as will become apparent from the following.

The sound of a bell is like certain pictures by Pablo Picasso: Many of
the objects included are familiar, but they are not found quite at those
places where they were supposed to be. The objects included in the sound
of a bell are its part tones, and the intervals between them. The part
tones of a typical contemporary church bell are by convention intended
to come as close as possible to the frequency ratios 1:2:2.4:3:4:5:6:8
(followed by some higher, less well defined part tones). Even if those
frequencies were exactly in the above ratios, the part-tone pattern
would not quite correspond to what the auditory system "supposes". While
some features of the pattern are familiar, as one can find two harmonic
series (1:2:3:4:5:6:8, and 2:4:6:8), other features are not quite as they
were "supposed to be": The minor third (2.4, with respect to 2) does neither
fit into the harmonic series (1:2:3:4:5:6:8) nor into the series (2:4:6:8).
And the fifth (3, with respect to 2) does not fit into the series (2:4:6:8).
Even if instead of the minor third a bell is cast with a major
third (2.5 instead of 2.4), the discrepancy remains. In reality, the part
tone frequencies may depart considerably (e.g., by a few percent) from
the above ratios - which provides for additional unexpected features of
the sound. The part tones of church bells cast in previous Centuries may
not even nearly follow the above pattern.

So it is apparent that both the auditory system and an acoustic theorist
should find it difficult to assign a pitch to such type of sound. The
theorist's problem is that neither from the time signal nor from the part-tone
spectrum one can "read" a period or a fundamental frequency, respectively,
that would tell the pitch. By contrast, the auditory system does not seem
to have much difficulty in assigning a strike tone to the sound, at least
for bells whose part tones largely follow the above pattern. This is why
through decades the strike note of bells has been regarded as a kind of
acoustic paradox.

The explanation is that the strike note indeed is a pitch,
namely, the most prominent one of the multiple pitches that are elicited
by the bell sound. The seeming paradox that one can assign one
pitch to a sound that actually has several pitches is resolved
by the notion that perception is hierarchically organized. An object that
at the bottom of the hierarchy appears just as a collection of multiple
elements (spectral pitches) may on a higher level of the hierarchy be
represented by just one perceptual object, i.e., a kind of Gestalt
(virtual pitch). The theory of virtual
pitcha priori is based on this hierarchical concept, and this
is why that theory explains the strike note of bells.

Grossly, the explanation is as follows. Any collection of simultaneous
part tones elicits a number of pitches which are both of the spectral
and virtual type. The relative prominence of all these pitches depends
in a fairly complicated manner on the frequencies and amplitudes of the
part tones. The most prominent pitch then is apprehended as the strike
note. Whether that pitch is of the virtual or spectral type, depends on
the frequencies (not just frequency ratios) and amplitudes of the part
tones.

This suggests that the strike note - in the sense of a musical pitch
- is by far not as pronounced and well defined as that of the conventional
tones of music, i.e., harmonic complex tones. That this indeed is true
becomes immediately apparent when one listens to music played on a carillon
(Glockenspiel). It is one of the main appeals of carillons that
recognition of a familiar melody requires some extra effort of auditory
analysis, and that in polyphonic music played on a carillon there occur
strange (dis)harmonies that can not - or not easily - be created by conventional
music instruments.

A most important implication of the aforementioned explanation of the
strike note is, that the theory accounts for the influence of the bell's
size, i.e., of the absolute part-tone frequencies. Assume, e.g., that
the part-tone pattern is of the aforementioned "ideal" type, and that
the bell is fairly big, such that the first part tone's frequency is 100
Hz. Then the first four part tones are below the so-called dominant
frequency region of the ear, which means that they hardly contribute
to the holistic pitch percept. The next part tones (400, 500, 600, 800
Hz), however, are well in the dominant region and will therefore be employed
by the auditory inference system of virtual pitch. The auditory inference
system will "conclude" that these part tones either are the 4th, 5th,
6th, and 8th harmonics of 100 Hz or that they are a mixture of the 2nd,
3rd, and 4th harmonics of 200 Hz with an extra 500-Hz tone that does not
count. As a result, both a virtual pitch corresponding to 100 Hz and another,
corresponding to 200 Hz will be signaled. As these two pitches are octave
equivalent, the musical pitch category is well defined and it is safe
to predict that the strike note will correspond to about the note G. The
octave region of the pertinent pitch is ambiguous, i.e., the strike note
may be heard both as G2 and G3, although it turns out that often G3 is
somewhat more pronounced. Note that to get this result, it is not required
that the part tones at and below 300 Hz are present or that they have
the correct frequencies. In this example, the strike-note pitch is of
the virtual-pitch type.

Now consider the case of a medium-sized bell of which the first part
tone of the series has the frequency of 500 Hz. In this case essentially
the first four part tones (500, 1000, 1200, 1500 Hz) must be considered,
because the higher ones are above the dominant frequency region. If the
500-Hz component is quite intense, it may alone suffice to determine the
pitch, such that the latter is a spectral pitch corresponding to 500 Hz.
If it the first part tone is not that strong, a virtual pitch will dominate
that is "deduced" from the 500, 1000, and 1500 Hz components and which,
of course, corresponds to 500 Hz, as well.

For a small-sized bell, e.g., with the first part tone at 1000 Hz or
higher, it is the first part tone alone that determines the pitch of the
bell, i.e., the strike note, because the higher part tones are above the
ear's dominant frequency region. Thus for small, i.e. high-pitched bells
the strike note throughout corresponds to the spectral pitch of the first
part tone.

From these considerations it is apparent that automatic, quantitative
prediction of strike notes is a challenge to any theory of pitch perception.
In a study of 17 and 137 historical church bells, respectively, we have
tested how well the predictions of the virtual-pitch theory agreed with
auditory pitch evaluation by listeners. The results were indeed satisfying
[46] ,
[65] ,
[104] p.
219-220.